Arithmetic volumes of indefinite ternary lattices with square free discriminant

不定三元非齐次二次型取值的有效密度

Effective density of values of indefinite ternary inhomogeneous quadratic forms

本文研究不定 $\mathbb{Q}$-迷向有理三元非齐次二次型在无理平移下的取值密度，给出范数有界整数向量落入任意目标区间的有效计数下界，指数由平移向量的丢番图性质决定，代数情形下可达到 $ν<1/8$。

On representation of an integer as the sum of three squares and the ternary quadratic forms with the discriminants p^2, 16p^2

Let s(n) be the number of representations of n as the sum of three squares. We prove a remarkable new identity for s(p^2n)- ps(n) with p being an odd prime. This identity makes nontrivial use of ternary quadratic forms with discriminants p^2 and 16p^2. These forms are related by Watson's transformations. To prove this identity we employ the Siegel--Weil and the Smith--Minkowski product formulas.

虚二次域上的三元完美型

Ternary Perfect Forms Over Imaginary Quadratic Field

本文枚举了绝对判别式不超过91的虚二次域上的三元完美型，证明其组合类型数量有与判别式无关的上界，并分析了相关数据。

Large moments and extreme values of class numbers of indefinite binary quadratic forms

Let $h(d)$ be the class number of indefinite binary quadratic forms of discriminant $d$, and let $\varepsilon_d$ be the corresponding fundamental unit. In this paper, we obtain an asymptotic formula for the $k$-th moment of $h(d)$ over positive discriminants $d$ with $\varepsilon_d\leq x$, uniformly for real numbers $k$ in the range $0<k\leq (\log x)^{1-o(1)}$. This improves upon the work of Raulf, who obtained such an asymptotic for a fixed positive integer $k$. We also investigate the distribution of large values of $h(d)$ when the $d$'s are ordered according to the size of their fundamental…

A conjecture on the zeta functions of pairs of ternary quadratic forms

We consider the prehomogeneous vector space of pairs of ternary quadratic forms. For the lattice of pairs of integral ternary quadratic forms and its dual lattice, there are six zeta functions associated with the the prehomogeneous vector space. We present a conjecture which states that there are simple relations among the six zeta functions. We prove that the coefficients coincide on fundamental discriminants.

Margulis函数的矩与不定三元二次型

Moments of Margulis functions and indefinite ternary quadratic forms

本文通过建立Margulis α-函数沿单幂轨道扩张平移的λ-矩一致界，并引入新高度函数刻画经典Margulis不等式的失效，从而证明了不定三元二次型Oppenheim猜想的定量版本。

Ternary Quadratic Forms, Modular Equations and Certain Positivity Conjectures

We show that many of Ramanujan's modular equations of degree 3 can be interpreted in terms of integral ternary quadratic forms. This way we establish that for any n in N |{n= x(x+1)/2 + y^2 +z^2 : x,y,z in Z}| >= |{n= x(x+1)/2 + 3y^2 +3z^2: x,y,zin Z}|, just to mention one among many similar positive results of this type. In particular, we prove the recent conjecture of H. Yesilyurt and the first author stating that for any n in N |{n= x(x+1)/2 + y^2 +z^2 : x,y,z in Z}| >= |{n= x(x+1)/2 + 7y^2 + 7z^2: x,y,z in Z}|. We prove a variety of identities for certain ternary forms with discriminants 1…

Ternary arithmetic, factorization, and the class number one problem

Ordinary binary multiplication of natural numbers can be generalized in a non-trivial way to a ternary operation by considering discrete volumes of lattice hexagons. With this operation, a natural notion of `3-primality' -- primality with respect to ternary multiplication -- is defined, and it turns out that there are very few 3-primes. They correspond to imaginary quadratic fields $\mathbb{Q}(\sqrt{-n})$, $n>0$, with odd discriminant and whose ring of integers admits unique factorization. We also describe how to determine representations of numbers as ternary products and related algorithms f…

Explicit Class number formulas for Siegel--Weil averages of ternary quadratic forms

In this paper, we investigate the interplay between positive-definite integral ternary quadratic forms and class numbers. We generalize a result of Jones relating the theta function for the genus of a quadratic form to the Hurwitz class numbers, obtaining an asymptotic formula (with a main term and error term away from finitely many bad square classes $t_j\mathbb{Z}^2$) relating the number of lattices points in a quadratic space of a given norm with a sum of class numbers related to that norm and the squarefree part of the discriminant of the quadratic form on this lattice.

三元二次型的定量强逼近 I

Quantitative strong approximation for ternary quadratic forms I

本文研究不定三元整系数二次型在局部条件下的定量强逼近，利用 Heath-Brown 的 δ 圆法得到含次级项、且允许 m 随 B 增长的整数解计数渐近公式。

Asymptotic formulas for class number sums of indefinite binary quadratic forms in arithmetic progressions

It is known that there is a one-to-one correspondence between equivalence classes of primitive indefinite binary quadratic forms and primitive hyperbolic conjugacy classes of the modular group. Due to such a correspondence, Sarnak obtained the asymptotic formula for the class number sum in order of the fundamental unit by using the prime geodesic theorem for the modular group. In the present paper, we propose asymptotic formulas of the class number sums over discriminants in arithmetic progressions. Since there are relations between the arithmetic properties of the discriminants and the conjug…

A quantitative Oppenheim Theorem for generic diagonal quadratic forms

We establish a quantitative version of Oppenheim's conjecture for generic ternary indefinite quadratic forms using an analytic number theory approach. The statements come with power gains and in some cases are essentially optimal

超奇异椭圆曲线的自同态环与三元二次型

Endomorphism Rings of Supersingular Elliptic Curves and Ternary Quadratic Forms

本文研究了超奇异椭圆曲线的自同态环，提出一种通过求解有限域平方根，从 Eichler order 高效计算 $B_{p,\infty}$ 中极大阶的算法，并给出了核理想右阶的基。

Hasse Principle Violations for Atkin-Lehner Twists of Shimura Curves

Let $D > 546$ be the discriminant of an indefinite rational quaternion algebra. We show that there are infinitely many imaginary quadratic fields $l/\mathbb Q$ such that the twist of the Shimura curve $X^D$ by the main Atkin-Lehner involution $w_D$ and $l/\mathbb Q$ violates the Hasse Principle over $\mathbb Q$.

CM-points and Lattice counting on arithmetic compact Riemann surfaces

Let $X(D,1) =Γ(D,1) \backslash \mathbb{H}$ denote the Shimura curve of level $N=1$ arising from an indefinite quaternion algebra of fixed discriminant $D$. We study the discrete average of the error term in the hyperbolic circle problem over Heegner points of discriminant $d <0$ on $X(D,1)$ as $d \to -\infty$. We prove that if $|d|$ is sufficiently large compared to the radius $r \approx \log X$ of the circle, we can improve on the classical $O(X^{2/3})$-bound of Selberg. Our result extends the result of Petridis and Risager for the modular surface to arithmetic compact Riemann surfaces.

A generalization of Watson transformation and representations of ternary quadratic forms

Let $L$ be a positive definite (non-classic) ternary $\z$-lattice and let $p$ be a prime such that a $\frac 12\z_p$-modular component of $L_p$ is nonzero isotropic and $4\cdot dL$ is not divisible by $p$. For a nonnegative integer $m$, let $\mathcal G_{L,p}(m)$ be the genus with discriminant $p^m\cdot dL$ on the quadratic space $L^{p^m}\otimes \q$ such that for each lattice $T \in \mathcal G_{L,p}(m)$, a $\frac 12\z_p$-modular component of $T_p$ is nonzero isotropic, and $T_q$ is isometric to $(L^{p^m})_q$ for any prime $q$ different from $p$. Let $r(n,M)$ be the number of representations of a…

Kitaoka 猜想与平方和

Kitaoka's Conjecture and sums of squares

本文建立了全实域上三元经典万有二次型存在性与2的所有全正倍数为平方和之间的联系，并证明了Kitaoka猜想对奇判别域成立。

Shimura correspondence for level $p^2$ and the central values of $L$-series II

Given a Hecke eigenform $f$ of weight $2$ and square-free level $N$, by the work of Kohnen, there is a unique weight $3/2$ modular form of level $4N$ mapping to $f$ under the Shimura correspondence. Furthermore, by the work of Waldspurger the Fourier coefficients of such a form are related to the quadratic twists of the form $f$. Gross gave a construction of the half integral weight form when $N$ is prime, and such construction was later generalized to square-free levels. However, in the non-square free case, the situation is more complicated since the natural construction is vacuous. The prob…